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A235189 Number of ways to write n = (1 + (n mod 2))*p + q with p < n/2 such that p, q and prime(p) - p + 1 are all prime. 9
0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 4, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 4, 2, 2, 4, 4, 1, 3, 2, 3, 2, 3, 3, 4, 3, 4, 3, 3, 3, 5, 2, 4, 4, 2, 2, 6, 2, 2, 4, 1, 1, 5, 4, 5, 4, 4, 2, 4, 3, 3, 3, 4, 4, 5, 4, 5, 4, 3, 2, 4, 2, 3, 6, 5, 3, 6, 3, 5, 5, 2, 3, 9, 3, 3, 5, 3, 1, 6, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

Conjecture: a(n) > 0 for all n > 6.

This implies both Goldbach's conjecture (A045917) and Lemoine's conjecture (A046927). For primes p with prime(p) - p + 1 also prime, see A234695.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

EXAMPLE

a(10) = 1 since 10 = 3 + 7 with 3, 7 and prime(3) - 3 + 1 = 3 all prime.

a(28) = 1 since 28 = 5 + 23 with 5, 23 and prime(5) - 4 = 7 all prime.

a(61) = 1 since 61 = 2*7 + 47 with 7, 47 and prime(7) - 6 = 11 all prime.

a(98) = 1 since 98 = 31 + 67 with 31, 67 and prime(31) - 30 = 97 all prime.

MATHEMATICA

p[n_]:=PrimeQ[Prime[n]-n+1]

a[n_]:=Sum[If[p[Prime[k]]&&PrimeQ[n-(1+Mod[n, 2])*Prime[k]], 1, 0], {k, 1, PrimePi[(n-1)/2]}]

Table[a[n], {n, 1, 100}]

CROSSREFS

Cf. A000040, A045917, A046927, A234694, A234695, A235187.

Sequence in context: A161056 A161260 A161285 * A308452 A103682 A023134

Adjacent sequences:  A235186 A235187 A235188 * A235190 A235191 A235192

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 04 2014

STATUS

approved

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Last modified August 6 12:33 EDT 2020. Contains 336246 sequences. (Running on oeis4.)